On the ergodicity properties of some adaptive MCMC algorithms

Annals of Applied Probability

Volumn 16, Issue 3, 2006, Pages 1462-1505

  Andrieu, Christophe ^a   Moulines, Éric ^b  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b TELECOM PARISTECH   (France)

Author keywords

Adaptive Markov chain Monte Carlo; Martingale; Metropolis Hastings algorithm; Poisson method; Randomly varying truncation; Self tuning algorithm; State dependent noise; Stochastic approximation

Indexed keywords

EID: 33750512542     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051606000000286     Document Type: Article

References (33)

1. ANDRIEU, C. (2004). Discussion on the meeting on statistical approaches to inverse problems, December 2003. J. R. Statist. Soc. Ser. B Stat. Methodol. 66 633-634.
2. ANDRIEU, C. and MOULINES, E. (2003). On the ergodicity property of some adaptive MCMC algorithms. Technical report, Univ. Bristol.
3. ANDRIEU, C., MOULINES, E. and PRIOURET, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283-312. MR2177157
4. ANDRIEU, C. and ROBERT, C. (2001). Controlled MCMC for optimal sampling. Cahiers du Cérémade0125.
5. ARCIDIACONO, P. and JONES, J. B. (2003). Finite mixture distributions, sequential likelihood and the EM algorithm. Econometrica 71 933-946. MR1983232
6. ATCHADÉ, Y. and ROSENTHAL, J. (2003). On adaptive Markov chain Monte Carlo algorithms. Technical report.
7. BARTUSEK, J. (2000). Stochastic approximation and optimization of Markov chains. Ph.D. thesis.
8. BAXENDALE, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700-738. MR2114987
9. BENVENISTE, A., MÉTIVIER, M. and PRIOURET, P. (1990). Adaptive Algorithms and Stochastic Approximations. Springer, New York.
10. BIRNBAUM, Z. W. and MARSHALL, A. W. (1961). Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32 687-703. MR0148106
11. CHEN, H., Guo, L. and GAO, A. (1988). Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 217-231. MR0931029
12. CHEN, H. and ZHU, Y.-M. (1986). Stochastic approximation procedures with randomly varying truncations. Sci. Sinica Ser. A 29 914-926. MR0869196
13. DUFLO, M. (1997). Random Iterative Systems. Springer, New York. MR1485774
14. DVORETZKY, A. (1972). Asymptotic normality for sums of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability II 513-535. Univ. California Press, Berkeley. MR0415728
15. GASEMYR, J. (2003). On an adaptive version of the Metropolis-Hastings with independent proposal distribution. Scand. J. Statist. 30 159-173. MR1965100
16. GELMAN, A., ROBERTS, G. and GILKS, W. (1995). Efficient Metropolis jumping rules. In Bayesian Statistics 5 (J. O. Berger, J. M. Bernardo, A. P. Dawid and A. F. M. Smith, eds.) 599-608. Oxford University Press, New York.
17. GELMAN, A. and RUBIN, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 473-483.
18. GLYNN, P. W. and MEYN, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 24 916-931. MR1404536
19. HAARIG, H., SAKSMAN, E. and TAMMINEN, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. MR1828504
20. HALL, P. and HEYDE, C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. MR0624435
21. HOLDEN, L. (1998). Adaptive chains. Technical report, Norwegian Computing Center.
22. JARNER, S. and HANSEN, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. MR1731030
23. KUSHNER, H. and YIN, G. (1997). Stochastic Approximation Algorithms and Applications. Springer, New York. MR1453116
24. MCLEISH, D. (1975). A maximal inequality and dependent strong laws. Ann. Probab. 3 829-839. MR0400382
25. MENG, X. L. and VAN DYK, D. (1997). The EM algorithm - an old folk song sung to a fast new tune. J. R. Statist. Soc. Ser B Stat. Methodol. 59 511-567. MR1452025
26. MENGERSEN, K. and TWEEDIE, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121. MR1389882
27. METROPOLIS, N., ROSENBLUTH, A., ROSENBLUTH, M., TELLER, A. and TELLER, M. (1953). Equations of state calculations by fast computing machines. J. Chemical Physics 21 1087-1091.
28. MEYN, S. and TWEEDIE, R. (1993). Markov Chains and Stochastic Stability. Springer, London. MR1287609
29. MEYN, S. and TWEEDIE, R. (1994). Computable bounds for convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011. MR1304770
30. NUMMELIN, E. (1991). On the Poisson equation in the potential theory of a single kernel. Math. Scand. 68 59-82. MR1124820
31. ROBERTS, G. and TWEEDIE, R. (1996). Geometric convergence and central limit theorem for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95-110. MR1399158
32. TITTERINGTON, D., SMITH, A. F. M. and MAKOV, U. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley, Chichester. MR0838090
33. Wu, C. (1983). On the convergence properties of the EM algorithm. Ann. Statist. 11 95-103. MR0684867